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| Mirrors > Home > MPE Home > Th. List > nnssn0s | Structured version Visualization version GIF version | ||
| Description: The positive surreal integers are a subset of the non-negative surreal integers. (Contributed by Scott Fenton, 17-Mar-2025.) | 
| Ref | Expression | 
|---|---|
| nnssn0s | ⊢ ℕs ⊆ ℕ0s | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-nns 28322 | . 2 ⊢ ℕs = (ℕ0s ∖ { 0s }) | |
| 2 | difss 4135 | . 2 ⊢ (ℕ0s ∖ { 0s }) ⊆ ℕ0s | |
| 3 | 1, 2 | eqsstri 4029 | 1 ⊢ ℕs ⊆ ℕ0s | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∖ cdif 3947 ⊆ wss 3950 {csn 4625 0s c0s 27868 ℕ0scnn0s 28319 ℕscnns 28320 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1542 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-v 3481 df-dif 3953 df-ss 3967 df-nns 28322 | 
| This theorem is referenced by: nnssno 28328 nnn0s 28333 nnn0sd 28334 nnsgt0 28343 | 
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